Kinematics
of Special Relativity| Kinematics
of Special Relativity |
General Concepts > s.a. lorentz group;
spacetime models; time;
[Reference
Frame].
* Interval: A useful expression is ds2 = –c2 t1t2,
where... @ Ohanian & Ruffini 94, p 70.
* Boost: A transformation
of coordinates given by t' = t cosh u +
x sinh u, x' = t sinh u + x cosh
u, for a boost along the x-axis (with tanh u = 
);
The coordinates y and z in
this case are unchanged,
and the boost
vector field is
v:= x 
/t
+ t /x ;
In general relativity, the corresponding conserved quantity (center of energy, > see conserved quantities) for asymptotically flat spacetimes is
M(N) = 
infty [N(a qbc –
b qac)
– N,a (qbc –
fbc) + N,b (qac –
fac)] qacqbd
dSd .
* Time dilation: T = 
T0;
Argue with bouncing photon clock; First tested in 1938 to within 1% of predictions
in an experiment by Herbert Ives and G R Stilwell; Verified with cosmic
ray muon decays and with GPS tests; 2007, best measurement by Gwinner et al
has accuracy quantified by a "Mansouri-Sexl parameter" of less than
8.4 
10–8.
* Length contraction: L = –1L0;
Argue based on time dilation and
L/T0 = L0/T (can
be illustrated with long car in garage); > s.a.
Lorentz-Fitzgerald Contraction.
* Velocities: Addition, u = (v1+v2)/(1+v1v2/c2) – neither
commutative nor associative due to the presence of Thomas precession; Four-velocity,
um
= dxm/d
.
* Doppler shift: Components
of vectors normal to are
invariant, so k' perp =
kperp; for other
ones,
k' 0 = (k0– · k)
, k' 1
= (k1–k0)
;
for light 
' = (1–cos
)
, tan' =
sin/[(cos–)]
.
@ Transformations: Brehme AJP(62)
[Lorentz and Galilei, geometrical]; Lévy-Leblond AJP(76),
Lévy AJP(07) [derivation].
@ Length contraction: Rindler AJP(61)
[paradox]; Nikolic AJP(99)phy/98 [accelerated
rod]; Pierce AJP(07)
[and lock and key paradox]; Redzic EJP(08) [interpretations].
@ Appearance of objects: Penrose PCPS(59)
[sphere]; Terrell PR(59);
in Van Bladel 84; Peres AJP(87);
Blackman EJP(98)ap/97;
Field AJP(00);
Morava a0804 [in terms of groups].
@ Time dilation: Bohm & Hiley AJP(85)
[and active interpretation of boosts]; Saathoff et al PRL(03)
[deviations < 2.2 ×
10–7]; Gwinner MPLA(05)
[tests, rev]; Harvey & Schücking PT(05)mar
[history, and gravitational time dilation]; news pw(07)nov [best measurement
yet].
Twin Paradox
* Idea: Identical twins
who are separated, undergo different accelerations, and are then reunited will
have aged differently; Not really a paradox, it is
a kinematical special relativity effect due to differences in proper times.
@ General references: Rodrigues & Rosa FP(89),
comment Sachs FP(89);
Harpaz EJP(90);
Debs & Redhead AJP(96)
[and simultaneity]; Ghosal et al FPL(05)
[and equivalence principle], gq/05;
Minguzzi AJP(05)
[explicit formula]; Grøn EJP(06);
Kohler FPL(06)
[instantaneous turnaround as limit]; Grandou & Rubin a0704-AJP
[as
a consequence of
causality].
@ Variations: Price & Gruber AJP(96);
Dray AJP(90),
Barrow & Levin
PRA(01)gq [closed
universe]; Cranor et al AJP(00),
Wortel et al AJP(07)gq [circular];
Iorio FPL(05);
Bansar et al gq/05 [with
a compact dimension]; Abramowicz et al PRA(07)
[on the photon sphere]; Field a0806.
Accelerated Observers > s.a. Acceleration: modified
electrodynamics; rotations.
@ General references: Desloge & Philpott AJP(87);
Desloge AJP(89)
[spatial geometry]; Tartaglia & Ruggiero EJP(03)gq;
Mashhoon gq/03-in,
gq/03-in
[non-locality]; Pardy gq/03;
Alba & Lusanna
gq/03 [3+1
view].
@ Rotation, rotating frames: Weber AJP(97)
[disk]; Tartaglia phy/98/FP
[Ehrenfest
paradox];
Herrera NCB(00)gq/99;
Bashkov & Malakhaltsev gq/00, gq/01;
Klauber gq/01 [frequency
and wavelength of light]; Rodrigues & Sharif FP(01)
[and
Sagnac
effect];
Rizzi & Ruggiero
FP(02),
03; Peres gq/04/AJP;
Klauber FP(07)gq/06;
Herrera
mp/06 [and
Thomas precession]; > s.a. Sagnac
Effect; time.
@ Related topics: Gruber & Price AJP(97)
[t dilation]; Chicone & Mashhoon
CQG(04)gq [critical
speed c/21/2].
Related Topics > s.a. clocks; Gyrogroup; Pythagorean
Theorem; Rapidity;
Rigid Body; Superluminal
Motion; tachyons.
@ Lorentz "rotations": Terrell PR(59);
Weisskopf PT(60)#9; in
Taylor & Wheeler 92, 92-93;
@ Simultaneity: Havas GRG(87);
Sarkar & Stachel PhSc(99);
Giulini gq/00;
Mamone FP(01);
Janis in(02);
Scherr et al AJP(02)
[student beliefs]; Bolós et
al
IJTP(02)gq/05 [spacelike
vs lightlike]; Minguzzi gq/05-in
[in special and general relativity]; > s.a. epr
paradox.
@ Velocity: Lévy-Leblond AJP(80); Mermin AJP(83), 90 ch19 [addition];
Sen JMP(90)
[relative velocities]; Mathews AJP(05)
[and acceleration].
@ Other topics: Christodoulou & O'Murchadha CMP(81)
[boost problem in general relativity];
Coll & Morales IJTP(92)
[frames]; Mermin
AJP(97), AJP(98)
[spacetime diagrams]; Pierceaux phy/04-in
[Poincaré's
kinematics and light fronts]; Price AJP(05)gq [projectiles
and pendula]; Dryzek et al AJP(06)
[lab experiment on E2 = p2c2 + m2c4].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
25 jun 2008